Constraints and generalized coordinates

Start with a fixed length

Imagine one point at the end of an ideal rigid rod of length 2 m. Its other end is fixed at O, whose coordinates are x = 1 m and y = −1 m. The point and rod are confined to one fixed plane. We choose perpendicular axes with x positive to the right and y positive upward. The point can occupy different positions, but its distance from O must remain 2 m.

You could record the point by two Cartesian coordinates, x and y. Yet you cannot choose both freely. If you put it directly above O, its position must be (1 m, 1 m); placing it at (1 m, 0 m) would give a distance of only 1 m. A constraint is a restriction on the configurations or motions admitted by the model.

The calculation needs substitution, signed coordinates and circle geometry. Units, standards, and physical constants, Algebra, functions, graphs, and logarithms, and Geometry, trigonometry, and coordinate systems provide that published preparation. We introduce the coordinate and constraint ideas here; equations of motion and variational methods are not needed to read this lesson.

See the allowed configurations

The Pythagorean relation gives the fixed-length condition: (x − 1 m)² + (y + 1 m)² = (2 m)². Here x and y are lengths measured in the chosen world coordinate system. The two offsets are the point’s coordinates relative to O. Every term in this equation has unit m².

The allowed configurations form the circle shown in the diagram. Its center is O, not the world coordinate origin. A, B, C and D are alternative positions of the same point. The solid radius selects B; the dashed radius OA defines the reference. Measure the angle θ counterclockwise from OA; at B it is a right angle, π/2 rad. Neither radius is a force or velocity arrow.

One choice along the circle determines the point’s position. This model has one configuration degree of freedom, meaning one independent local coordinate is needed. The circle lies in a two-dimensional physical plane even though the constrained configuration needs only one coordinate. These are different counts.

Use one angle to locate the point

Use the angle θ defined in the diagram to locate the rod about O. A full turn is 2π rad. The radian identifies a plane angle; in the SI it is a special name for the unit one. Keeping rad visible distinguishes this coordinate from a length.

Write φ = θ/(1 rad), the dimensionless numerical angle in radians. The circle geometry gives x = 1 m + (2 m) cos φ; y = −1 m + (2 m) sin φ. Cosine and sine give the horizontal and vertical components of the unit-radius direction. Multiplying them by 2 m and adding O’s coordinates restores the world position.

For example, θ = π/2 rad gives cos φ = 0 and sin φ = 1, hence B = (1 m, 1 m). The identity cos² φ + sin² φ = 1 shows that this transformation satisfies the fixed-length condition for every angle, not just the four table entries.

A generalized coordinate is a variable chosen to describe a configuration through such a reconstruction rule. It may be a length, an angle or another suitable parameter. In this model we may take q = θ; the symbol q does not require a Cartesian direction or a metre unit. Each coordinate still needs a meaning and a domain.

Check completeness and coordinate domains

A proposed coordinate must distinguish the configurations it is meant to cover. The value x = 1 m does not distinguish B from D: their y values are 1 m and −3 m. Thus x alone is incomplete for the whole circle unless branch information is supplied.

On the upper open semicircle, one can instead use −1 m < x < 3 m and reconstruct y = −1 m + √[(2 m)² − (x − 1 m)²]. The positive square root selects the upper branch. This is a local coordinate choice on a stated region; the lower branch needs the negative square root. At the left and right endpoints this particular description is not a smooth coordinate, so another local choice is useful.

Angles also need a convention. Choosing 0 ≤ θ < 2π rad represents each circle position once, but introduces a seam at A: positions just below a full turn are physically near positions just above zero. A local angle interval crossing A can use negative angles as well as positive ones. One real interval does not supply a single smooth coordinate chart for the entire closed circle.

For configurations, θ = 0 rad and θ = 2π rad give the same spatial position. A continuous unwrapped angle history can retain the number of complete turns, but no such history has been supplied here. Coordinate representatives and the accumulated motion are different information.

Count independent restrictions

Consider a different supplied model: two sliders on separate straight rails have position coordinates x₁ and x₂, with 0 ≤ x₁ ≤ 5 m, 0 ≤ x₂ ≤ 5 m and x₁ + x₂ = 5 m. The rails and their coordinate origins are fixed. This exact coupling rule is stipulated; we are not deriving a real cable or a force law.

Choose q = x₁. Then x₂ = 5 m − q, with 0 ≤ q ≤ 5 m. For q = 2 m, the two coordinates are 2 m and 3 m. One independent coordinate specifies the whole configuration, although two position numbers can be written down. At either interval endpoint, allowed small changes are one-sided.

Adding 2x₁ + 2x₂ = 10 m does not remove another choice: it is twice the same equation. Adding x₁ = 2 m does remove the remaining choice, selecting the single configuration (2 m, 3 m). Count independent restrictions rather than the number of equations printed.

The familiar local rule is n − k coordinates for n position variables subject to k independent smooth equality constraints at a regular configuration. Regular means that the restrictions remove the stated independent local directions without a singular collapse. It is a local result with conditions, not a rule for every inequality, corner or exceptional point.

Treat prescribed time as given

For a third model, a point lies on a horizontal rail whose height is externally prescribed. During 0 ≤ t ≤ 4 s, supply y = 1 m + (0.5 m/s)t. At t = 0, 2 and 4 s, the rail heights are 1 m, 2 m and 3 m. The ideal rail has no horizontal end stops, so x may be any finite real length.

At a given time choose q = x. The reconstruction is x = q together with the supplied y(t). At t = 2 s and q = −1 m, the configuration is (−1 m, 2 m). Time is a given parameter in this description, not a second independently unknown position coordinate.

A position constraint that can be written using coordinates and possibly time is called holonomic. The fixed circle and slider coupling are time-independent holonomic constraints; the prescribed moving rail is time-dependent. If the support height were freely movable instead of prescribed, it would be a different model and could require another independent coordinate.

Notice where the simple count fails

Keep the model’s bounds as well as its equalities. In the slider example, q = 6 m would give x₂ = −1 m and violate the nonnegative-position domain. A formula that produces a number is not enough to make that number an admitted configuration.

A singular case also changes counting. Replacing the circle’s 2 m radius by zero gives (x − 1 m)² + (y + 1 m)² = 0 m². Both squared terms are nonnegative, so only O is possible. There are zero free configuration coordinates, even though only one equation is written. The regularity condition behind the n − k rule has failed.

Some restrictions are stated in terms of velocities. A velocity relation can sometimes be integrated into a position relation, while other velocity restrictions cannot be reduced that way. The latter are nonholonomic. Do not classify a constraint merely because its first written form contains a velocity, or subtract every velocity relation as if it removed a configuration coordinate.

Separate configuration from motion

The configuration is the system’s allowed position arrangement. Our circle equation does not tell us which allowed position occurs at a later time, how fast the point moves, or what forces maintain the restriction. Many different angle histories can satisfy the same fixed length. Choosing a convenient coordinate organizes the question; it does not solve the dynamics.

The assumptions determine what counts as a configuration. If the fixed-length point were free to move in three-dimensional space rather than our plane, its allowed positions would form a sphere and would require two local position coordinates at regular points. The imposed plane is essential to our one-angle circle model.

A useful procedure is to state the objects and restrictions, choose independent coordinates with domains and units, reconstruct the original positions, and substitute back into every constraint. Then add the motion and interaction information needed for the next question. Virtual work and d’Alembert’s principle develops how such restricted configurations enter dynamics.

Worked examples

Try it yourself

1. In the circle model, compare the proposed positions (1 m, 1 m) and (1 m, 0 m). Which is allowed?

   Show the answer and reasoning

   (1 m, 1 m) is allowed; (1 m, 0 m) is not.

   Their distances from O = (1 m, −1 m) are 2 m and 1 m respectively. The fixed-length condition requires 2 m, rather than merely a point inside the circle.

2. Find the positions for θ = π/2 rad and θ = 3π/2 rad. Does x alone identify which one is present?

   Show the answer and reasoning

   The positions are (1 m, 1 m) and (1 m, −3 m). Both have x = 1 m, so x alone is insufficient on the whole circle.

   Sine changes from +1 to −1 while cosine is zero in both cases. The angle, or x together with branch information, distinguishes the configurations.

3. Does θ = 2π rad give a new spatial configuration compared with θ = 0 rad? Which representative is used by 0 ≤ θ < 2π rad?

   Show the answer and reasoning

   They give the same configuration A = (3 m, −1 m). The stated representative range uses 0 rad and excludes 2π rad.

   A full turn restores the same direction from O. An unwrapped motion history could count that turn, but one configuration reading does not establish that it occurred.

4. In the sum-5 m slider model, find x₂ for q = x₁ = 3.5 m. Would q = 6 m be allowed?

   Show the answer and reasoning

   For q = 3.5 m, x₂ = 1.5 m. The value q = 6 m is outside the allowed domain.

   Subtract q from 5 m and check both nonnegative bounds. The second value would give x₂ = −1 m, so satisfying the sum alone is not enough.

5. At t = 4 s on the moving rail, choose q = x = 2 m. Give the configuration and its number of independent position coordinates at that fixed time.

   Show the answer and reasoning

   The configuration is (2 m, 3 m), with one independent position coordinate at the given time.

   The prescribed height is 1 m + (0.5 m/s)(4 s) = 3 m. x remains the one freely chosen coordinate; t is supplied rather than another unknown configuration variable.

6. Keep the pivot and positive 2 m length but remove the fixed-plane restriction. For one point in three-dimensional space, how many local configuration coordinates are needed at a regular point?

   Show the answer and reasoning

   Two local configuration coordinates are needed.

   The allowed positions form a sphere: three Cartesian position variables obey one regular fixed-length equality. The original plane restriction was what reduced the sphere to a one-coordinate circle.

What to carry forward

• A constraint restricts admitted configurations or motions; it does not by itself prescribe the motion history or forces.
• For the fixed-plane, fixed-length point, one angle reconstructs two related Cartesian coordinates.
• Generalized coordinates need complete reconstruction rules, units and domains; branches and chart seams matter.
• Count independent regular restrictions locally. Redundant equations, singular cases and inequality boundaries need separate checks.
• An externally prescribed time-dependent support is different from an additional freely movable part of the system.